825 



S(Y + iv, 0) + S(Y - h, 0) - 2S(x, 0)]} . 



(59) 



Now when we expand the exponent in powers of y the lin- 

 ear terms vanish, so that we have 



X exp{2!9[S(x, Y) + S(Y, 0) - S(x, 0), , 

 x| d'y p(y,Y) explUQy,yiA,j(Y)] , 

 where we define 



>lu(Y) = ^^[S(x,Y) + S(y,0)|. 



(60) 



(61) 



Evaluation of the integral on rf'Y by stationary phase 

 again selects as the stationary phase path the unper- 

 turbed ray joining to x. The integral on rf'y can then 

 be done by introducing the Fourier transform p(k, Y) as 

 in Eq. (58). Finally we obtain 



(X(x)2> = -|-j [ ds{ d'Kis)p(K(s)Y(s)) 

 4ir Jo J 



ye7cp[ii/g)k.,(s)hi{s)A-Hns)),,] . 



(62) 



The notation is as in Eq. (57), and the result is again 

 in complete analogy to the homogeneous case. 



Most of the comments we made in Sec. I concerning 

 the results in the homogeneous background apply here 

 as well. The expression for ( lA'l^) is again just that 

 obtained in the geometrical optics approximation, but 

 that for (X^) is not. Geometrical optics for (X^> is 

 valid provided that 



(l/q)k,,{s)h,{s)A-HY{s)), 



«1 



(63) 



This is the analog of the Fresnel condition. 



III. CHANNELED OCEAN 



Let us next apply our general results, Eqs. (57) and 

 (62), to the specific case of a channeled ocean with its 

 associated cylindrical symmetry. We shall choose the 

 z axis as the vertical with z positive upward. We shall 

 choose the unperturbed ray of interest to lie in the x, z 

 plane, and we shall confine ourselves to situations in 

 which the source and receiver lie at the same depth. 

 Thus the source is at the point (x, y, z) = (0, 0, 0) and the 

 receiver is at the point (R, 0, 0) where R is the range. 

 The unperturbed ray path joining these two points will 

 be denoted z(x); thus z(0) = 0, z(i?) = and 



6(x) 



-"■'m 



is the angle the ray makes with the horizontal at the 

 point X. The element of path length along the ray is 

 then given by 



ds = [l+tiin^e(x)Y'^dx. 



The expressions (57) and (60) for ( 1XI^> and (X^) 

 both involve integrals of the Fourier components of the 

 correlation function over a plane perpendicular to the 

 ray path at each point along the ray. With our geome- 

 try, the wave number k(s) perpendicular to the ray has 



X, y, and z components 



and the element of surface area perpendicular to the 

 ray is 



d^K(s) = dk^dk,/cose . 

 Thus Eq. (57) becomes 



<|x|'>=f^ f dx sec^e f dkA dk. 



xp{[- k.tuneix), k^, k,], zix)) . (64) 



Now it turns out to be more convenient to express the 

 correlation function p in terms of the variables oi and j, 

 the frequency and mode number of the internal waves, 

 rather than the wave numbers k^ and k,. The transfor- 

 mation to these new coordinates is accomplished as fol- 

 lows: horizontal and vertical components of wave num- 

 ber 



*„ = (*^tan2e + y^)''% ky = k, 

 have the approximate dispersion relations [Eq. (92)] 



k„=JTrB-Ho}'-i^\J^'^/no , ky = ji< B'^ n/ n^ . 

 Then we note the definition 



j j^'p(% ^) = -^ ] k„ dk„ j dk.'piku, k„z) 



= S r du^F(ui,j■,^) , (65) 



where F(cl),;) is the spectrum of bC/C, and 



The internal wave vector k has an inclination ky /k^; 

 group velocity is at right angles, with inclination k„/ky, 

 and its component in the plane of sound propagation is 

 kjk^<k„/ky. Atcu=aJi, k„/ky = (Ji-uj\J'^/n = t3Ln6; 

 lesser frequencies have k,/ky<\.^n6, and are excluded 

 from consideration in the stationary phase approxima- 

 tion. 



Thus Eq. (64) may be replaced by 



( |x|^> = 2;r"V«o-Bf rfrsec^^I] j^M dui 



x(i/-o,\r"^F(w,j;z). (66) 



In this expression z, n,. and 9 are of course functions of 

 X, to be evaluated at each point along the unperturbed 

 ray as x varies from source to receiver. 



A similar transformation may be carried out for {X^), 

 the other quantity of interest. The only complication 

 here is that it is necessary to evaluate the matrix At), 

 introduced in Eq. (61), at each point along the ray. 

 The symmetry associated with a channeled ocean makes 

 this relatively easy to do, as follows. First, in the 

 horizontal plane the unperturbed sound speed is con- 

 stant. Hence the optical path between any two points 

 (ri.ViZi) and (xjyjZj) can be written 



SixzVzZz, XiyiZi) = X2 - :»ri + ( ya - J'i)V2(Ar2 - rj 



+ S'(*2Z3, ATiZi) , (67) 



219 



